Interacting Dirac liquid in three-dimensional semimetals

We study theoretically the properties of the interacting Dirac liquid, a novel three-dimensional many-body system which was recently experimentally realized and in which the electrons have a chiral linear relativistic dispersion and a mutual Coulomb interaction. We find that the “intrinsic” Dirac liquid, where the Fermi energy lies exactly at the nodes of the band dispersion, displays unusual Fermi liquid properties similar to graphene, whereas the “extrinsic” system with finite detuning or doping behaves as a standard Landau Fermi liquid. We present analytical and numerical results for the self-energy and spectral function based on both Hartree-Fock and the random phase approximation (RPA) theories and compute the quasiparticle lifetime, residue, and renormalized Fermi velocity of the extrinsic Dirac liquid. A full numerical calculation of the extrinsic RPA spectral function indicates that the Fermi liquid description breaks down for large-energy excitations. Furthermore, we find an additional plasmaron quasiparticle sideband in the spectral function which is discontinuous around the Fermi energy. Our predictions should be observable in ARPES and STM measurements.

Figure 1

Diagrammatic representation of the self-energy in the random phase approximation. Thin straight and wiggly lines denote the noninteracting Green's function $G_0$ and the bare Coulomb interaction $V_{\mathbf{q}}$, respectively. The dynamically screened RPA interaction (thick wiggly line) is obtained by a summation of the geometric series of Coulomb lines and free polarizability.

Figure 2

Full spectral function at $\alpha =0.15$. Clearly visible are the plasmaron sidebands, which have a discontinuous jump across the Fermi momentum. While around the Fermi surface we find a renormalized linear quasiparticle branch, this description breaks down at small momentum and negative frequency, where the quasiparticle cone hybridizes with the plasmaron branch. For comparison, we indicate the peak position of the noninteracting spectral function by white dashed lines. The increased slope of the quasiparticle band is a sign of the Fermi velocity renormalization. The insets show aspects of a typical 2D Dirac material spectral function with $\alpha =0.5$: (a) plasmaron-ring structure near the Dirac point, (b) the plasmaron branch wraps around the Fermi surface, reflecting the ungapped $\sqrt{q}$-2D plasmon mode.

Figure 3

Spectral function at fixed momentum $k/k_F=0.75,1,$ and $1.25$. There is a dominant peak associated with the quasiparticle and a plasmaron side peak. At $k_F$, the quasiparticle peak is a delta function, and the plasmaron excitations are gapped.

Figure 4

(a) Occupation number and (b) density of states. The blue line denotes the interacting result and the red line denotes the noninteracting DOS.